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DUALITY OF SEQUENCE SPACES OF INFINITE MATRICES
By
Mr. Suchat Samphavat
A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree
Master of Science Program in Mathematics
Department of Mathematics
Graduate School, Silpakorn University
Academic Year 2011
Copyright of Graduate School, Silpakorn University
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DUALITY OF SEQUENCE SPACES OF INFINITE MATRICES
By
Mr. Suchat Samphavat
A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree
Master of Science Program in Mathematics
Department of Mathematics
Graduate School, Silpakorn University
Academic Year 2011
Copyright of Graduate School, Silpakorn University
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The Graduate School, Silpakorn University has approved and accredited the Thesis title of “Duality of sequence spaces of infinite matrices” submitted by MR.Suchat Samphavat as a partial fulfillment of the requirements for the degree of Master of Science in Mathematics
….……..……........................................................... (Assistant Professor Panjai Tantatsanawong,Ph.D.) Dean of Graduate School ........../..................../..........
The Thesis Advisor Jitti Rakbud, Ph.D. The Thesis Examination Committee .................................................... Chairman (Somjate Chaiya, Ph.D.) ............/......................../.............. .................................................... Member (Assistant Professor Poom Kumam, Ph.D.) ............/......................../.............. .................................................... Member (Jitti Rakbud, Ph.D.) ............/......................../..............
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52305204 : MAJOR : MATHEMATICS KEY WORD : BANACH SPACE / DUALITY / DIRECT- SUM 1l SUCHAT SAMPHAVAT : DUALITY OF SEQUENCE SPACES OF INFINITE MATRICES. THESIS ADVISOR : JITTI RAKBUD, Ph.D.. 31 pp.
In this thesis, we define, for each 1 r , the set of infinite complex matrices as follows:
r
( ) ( ) 2
1 1: :
rr k kji jik k
a a B l .
We first show as a preliminary that equipped with the norm 1/
( ) ( )
1 1:
rrk k
ji jik kra a ,
the set is a Banach space. The main goal of this research is to decompose
the dual
r
r of as an direct- sum of its two closed subspaces by a way
analogous to the classical theorem of Dixmier on decomposing the dual
r 1l2B l of
2B l .
Department of Mathematics Graduate School, Silpakorn University Student's signature ........................................ Academic Year 2011 Thesis Advisor's signature ........................................
d
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52305204 : : / / 1l
: . : . . . 31 .
r
r1 ( ) ( ) 2
1 1: :
rr k kji jik k
a a B l
r
1/( ) ( )
1 1:
rrk k
ji jik kra a
r r
1l 2 r 2B l 2B l
.......................................... 2554
........................................ e
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ACKNOWLEDGEMENTS
This thesis has been completed by the involvement of people about whom I would like to mention here. I would like to thank Dr. Jitti Rakbud, my advisor, for his valuable suggestions and excellent advices throughout the study with great attention. I would like to thank Dr. Somjate Chaiya and Asst. Prof. Dr. Poom Kumam for their valuable comments and suggestions. Finally, I would like to thank the centre of excellence in mathematics, the commission on higher education, Thailand, for the financial support .
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TABLE OF CONTENTS Page Abstract in English.............................................................................................. d Abstract in Thai ................................................................................................... e Acknowledments.................................................................................................. f Chapter 1 Introduction and Preliminaries .............................................................. 1 2 Theoretical Background Banach Spaces……………………………………………………. .... 4 Spaces....................................................................................... 7 pl Hilbert Spaces………………………………................................... 8 Schatten - Classes..................................................................... 12 p 3 Duality of sequence spaces of infinite matrices Basic Results………….. ................................................................ 14 Duality……………………………………. ....................................... 18 4 Conclusion................................................................................................. 27 Bibliography…………………………………………………………………............... 30 Biography ............................................................................................................. 31
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Chapter 1
Introduction and Preliminaries
A beautiful decomposition of the dual B(l2)∗ of the Banach algebra B(l2) ofbounded linear operators on l2, under the usual multiplication, was established byDixmier (see [5] and [6]). He proved that every bounded linear functional f in B(l2)∗
can uniquely be decomposed as the sum f = g + h of two bounded linear functionalsh and g on B(l2) such that g is a trace functional defined associated with a trace classoperator T by g(S) = trace(ST ) for all S ∈ B(l2), and h vanishes on the ideal K(l2)of compact operators on l2. The most interesting part of the theorem of Dixmiermentioned above is that the norm of the decomposition f = g +h of each f in B(l2)∗
is additive, i.e., ‖f‖ = ‖g‖+ ‖h‖. A part of Schatten’s theorem (see [12]) states thatK(l2)∗ ∼= C1, where C1 denotes the class of all trace class operators on l2. From thetheorem of Schatten, it can easily be deduced that the space of all trace functionalson B(l2) is isometrically isomorphic to the dual K(l2)∗ of K(l2). Thus the theoremof Dixmier mentioned above can be symbolized as B(l2)∗ = K(l2)∗ ⊕1 K(l2)s, whereK(l2)s denotes the space of all linear functionals on B(l2) vanishing on K(l2), whichare called singular functionals on K(l2), and the notation “⊕1” is referred to as thel1 direct-sum.
Let 1 ≤ p, q, r < ∞. An infinite scalar matrix A = [ajk] is said to define
a linear operator from lp into lq if for every x = {xk}∞k=1 in lp the series∞∑
k=1
ajkxk
converges for all j, and the sequence Ax :=
{ ∞∑k=1
ajkxk
}∞
j=1
is a member of lq. If a
matrix A defines a linear operator from lp into lq, we then call the operator x �→ Axthe linear operator defined by A. In this case, it can be shown by the uniformboundedness principle that the linear operator defined by A is bounded. Let M(lp, lq)be the set of all infinite matrices which define linear operators from lp into lq. Foreach matrix A, we call A a bounded matrix and define ‖A‖ to be the norm of thelinear operator defined by A if A ∈ M(lp, lq) and call A an unbounded matrix anddefine ‖A‖ to be ∞ otherwise. It is well-known that M(lp, lq) is a Banach spaceunder the norm ‖·‖. Indeed, it coincides with the set of matrix representations ofall bounded linear operators from lp into lq with respect to the standard Schauderbases of lp and lq, which is isometrically isomorphic to the Banach space B(lp, lq) ofall bounded linear operators from lp into lq. A matrix A is called a compact matrixif the linear operator defined by A is a compact operator.
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For each matrix A = [aji] and positive integer n, let An� = [bji] be the matrixwith bji = aji for all 1 ≤ j, i ≤ n and bji = 0 otherwise, and let An� = [cji] bethe matrix with cji = aji for all j, i ≥ n and cji = 0 otherwise. The following arewell-known facts about infinite matrices which are useful for the research.
Theorem 1.1.
(1) If [aji] and [bji] are scalar matrices such that |aji| ≤ bji for all j, i, then ‖[aji]‖ ≤‖[|aji|]‖ ≤ ‖[bji]‖.
(2) A matrix A belongs to B(lp, lq) if and only if supn
‖An�‖ < ∞.
(3) For every matrix A, ‖An�‖ ↗ ‖A‖.(4) For each A ∈ B(l2) and positive integer n, ‖An� + An�‖ = max{‖An�‖ , ‖An�‖}.(5) A matrix A is compact as an operator on l2 if and only if ‖An� − A‖ → 0.
The Schur product or Hardamard product or entry-wise product of two scalarmatrices A = [ajk] and B = [bjk] having the same size is defined by the matrixA • B := [ajkbjk]. In [13], Schur proved that Banach space B(l2) is a commutativeBanach algebra (without identity) under the operator norm and the Schur productmultiplication. After that, Bennett extended in [1] the result of Schur referred toabove. He showed for each 1 ≤ p, q < ∞ that the Banach space B(lp, lq) under theSchur product operation is also a Banach algebra. These beautiful results of Bennettmotivated Chaisuriya and Ong [2] to study some classes of infinite matrices overBanach algebras with identity. In [2], for a fixed Banach algebra B with identityand 1 ≤ p, q, r < ∞, the authors defined the class Sr
p,q(B) of matrices A = [ajk] over
B such that the absolute Schur rth − power A[r] := [‖ajk‖r] defines a linear operatorfrom lp into lq. And then they proved that it is a Banach algebra under the theabsolute Schur r-norm defined by
‖|A|‖p,q,r =∥∥A[r]
∥∥1/r
and the Schur product, which is straightforwardly generalized to the setting of ma-trices over the Banach algebra B by using the multiplication in B. The authors alsoprovided a beautiful relationship, which follows from the results of Schur and Ben-nett mentioned above, between the algebra B(lp, lq) of all bounded operators form lp
into lq and the algebra Srp,q(C). They found that B(lp, lq) is contained in Sr
p,q(C) asa non-closed ideal for all r ≥ 2.
In [8], Livshits, Ong and Wang studied the duality of the absolute Schuralgebras Sr
2,2(C) by a way analogous to Dixmier’s theorem and Schatten’s theoremmentioned in the first paragraph. The authors defined the class Kr of infinite matricesA such that A[r] is compact as an operator on l2 for playing the role as the classK(l2) of all compact operators on l2. They also constructed a class Mr of infinitematrices for playing the role as the class C1 of all trace class operators, which isknown as the dual of K(l2). They obtained that (Kr)∗ ∼= Mr and that each bounded
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linear functional ϕ on Sr2,2(C) can uniquely be decomposed as the sum ϕ = ρ + ψ,
where ρ is determined by a unique matrix in Mr under a certain way and ψ isa singular functional on Kr. Furthermore, the decomposition ϕ = ρ + ψ satisfies‖ϕ‖ = ‖ρ‖+ ‖ψ‖. Schatten’s theorem also states that the trace class operators forma predual of B(l2). An analogue of this result on the setting of Livshits, Ong andWang: (Mr)∗ ∼= Sr
2,2(C), was also obtained.
From the beautiful result of Chaisuriya and Ong that the absolute Schuralgebra S2
2,2(C) contains B(l2) as a non-closed ideal, Rakbud and Ong defined threesequence spaces of matrices from S2
2,2(C) in [11] as follows:
Ob =
{{Ak}∞k=1 ⊆ S2
2,2(C) : the sequence
{n∑
k=1
A[2]k
}∞
n=1
is bounded in B(l2)
},
Oc =
{{Ak}∞k=1 ⊆ S2
2,2(C) : the sequence
{n∑
k=1
A[2]k
}∞
n=1
converges in B(l2)
},
and
Oκ =
{{[a
(k)ji
]}∞
k=1⊆ S2
2,2(C) : the matrix
[ ∞∑k=1
∣∣∣a(k)ji
∣∣∣2] ∈ K(l2)
}.
The authors obtained the inclusion relation among these three spaces as follows:Oκ � Oc � Ob. They defined naturally a norm on these three spaces by
‖|{Ak}∞k=1|‖ =
(sup
n
∥∥∥∥∥n∑
k=1
A[2]k
∥∥∥∥∥)1/2
and showed that all three sequence spaces equipped with this norm are Banach spaces.It was observed that because of the non-closedness of B(l2) in S2
2,2(C), the restrictionsof these sequence spaces to B(l2) are all not complete. The study on this paper wasmainly focused on the sequence spaces Oc and Oκ. The authors studied sequentialconvergence in these two sequence spaces and duality and preduality of Oκ.
From the idea of Rakbud and Ong referred to above, we obtain a way analo-gous to the classical sequence spaces lp to define sequence spaces of infinite matricesas follows. Let M∞ be the vector space of all infinite complex matrices. For each1 ≤ r < ∞, let
L r =
{{[a
(k)ji
]}∞
k=1⊆ M∞ :
[ ∞∑k=1
∣∣∣a(k)ji
∣∣∣r] ∈ B(l2)
}.
In this research, we study some elementary properties and provide some resultson duality of the sequence spaces L r. The main goal is to establish a decompositiontheorem for the dual space (L r)∗ of L r by a way analogous to the theorem ofDixmier mentioned in the first paragraph.
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Chapter 2
Theoretical Background
In this chapter, we provide some theoretical background which is necessaryfor the research.
Throughout this thesis, we let C and R denote the sets of all complex numbersand real numbers respectively.
2.1 Banach Spaces
Definition 2.1.1. [9] Let X be a vector space over a scalar field K (K = R or C).A norm on X is a real-valued function ‖·‖ on X satisfying the following properties:
(i) ‖x‖ ≥ 0;
(ii) ‖x‖ = 0 if and only if x = 0;
(iii) ‖αx‖ = |α| ‖x‖;(iv) ‖x + y‖ ≤ ‖x‖ + ‖y‖ (Triangle inequality),
where x and y are arbitrary vectors in X and α is any scalar in K. A normed spaceis a pair (X, ‖·‖) of a non-empty set X and a norm ‖·‖ on X. It may be sometimeswritten just X as a normed space by omitting the norm on X.
Definition 2.1.2. [9] A sequence {xn}∞n=1 in a normed space X is said to convergeor to be convergent if there is a point x in X satisfying the following property: forany ε > 0, there is a positive integer N such that
‖x − xn‖ < ε for all n ≥ N.
In this situation, we write limn→∞
xn = x, or simply xn → x and call x the limit of
{xn}∞n=1.
Definition 2.1.3. [9] A sequence {xn}∞n=1 in a normed space X is said to be boundedif there is a positive real number c such that ‖xn‖ ≤ c for all positive integer n.
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Definition 2.1.4. [9] A sequence {xn}∞n=1 in a normed space X is said to be a Cauchysequence in X if for any ε > 0, there is a positive integer N such that
‖xm − xn‖ < ε
for all m,n ≥ N . A normed space X is said to be a Banach space if it is completeunder the metric d defined by d(x, y) = ‖x − y‖ , that is, every Cauchy sequenceconverges to an element in X.
Definition 2.1.5. [9] Let X and Y be vector spaces over the same scalar field. Afunction T : X → Y is said to be a linear operator or linear function or lineartransformation if
T (αx1 + βx2) = αTx1 + βTx2
for every x1, x2 ∈ X and any scalars α and β.
Definition 2.1.6. [9] Let X and Y be normed spaces over the same scalar field. Alinear operator T : X → Y is said to be bounded if T (B) is bounded for all boundedsubsets B of X.
Definition 2.1.7. Let T be a linear operator from a normed space X into a normedspace Y . Then the range of T is denoted by ran T . We call the set {x ∈ X : Tx = 0}the kernel of T and denote by ker T .
Theorem 2.1.8. [9] Let T : X → Y be a linear operator from a normed space Xinto a normed space Y . Then the following are equivalent.
(1) T is bounded.
(2) T is continuous.
(3) There is a constant M > 0 such that ‖Tx‖ ≤ M ‖x‖ for all x ∈ X.
Let B(X,Y ) be the set of all bounded linear operators from a normed spaceX into a normed space Y . We denote B(X, X) by just B(X).
Definition 2.1.9. [9] Let X and Y be normed spaces. For each T in B(X, Y ), thenorm or operator norm ‖T‖ of T is the nonnegative real number sup{‖Tx‖ : x ∈X, ‖x‖ ≤ 1}. The operator norm on B(X,Y ) is the map T �→ ‖T‖.
From Theorem 2.1.8, the following corollary is immediately obtained.
Corollary 2.1.10. [9] If T is a bounded linear operator from a normed space X intoa normed space Y , then ‖Tx‖ ≤ ‖T‖ ‖x‖ for all x in X. Furthermore, the number‖T‖ is the smallest nonnegative real number M such that ‖Tx‖ ≤ M ‖x‖ for allx ∈ X.
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Definition 2.1.11. [9] Let T be a linear operator from a normed space X onto anormed space Y . The operator T is an isometric isomorphism if ‖T (x)‖ = ‖x‖whenever x ∈ X.
Notice that the condition ‖T (x)‖ = ‖x‖ for all x ∈ X implies T is an one-to-one function.
Theorem 2.1.12. [9] If X is a normed space and Y is a Banach space, then the setB(X, Y ) equipped with the operator norm is a Banach space.
Theorem 2.1.13. [9] (The Uniform Boundedness Principle) Let F be a nonemptyfamily of bounded linear operators from a Banach space X into a normed space Y .If sup {‖Tx‖ : T ∈ F} is finite for each x in X, then sup{‖T‖ : T ∈ F} is finite.
Definition 2.1.14. [9] A normed space X is said to be the direct sum of its twosubspaces Y and Z, written by X = Y ⊕Z, if each x ∈ X has a unique representationof the form x = y + z, where y ∈ Y and z ∈ Z. If, in addition, the condition‖x‖ = ‖y‖ + ‖z‖ is satisfied for all x ∈ X, we say specifically that X is the l1
direct-sum of Y and Z and write X = Y ⊕1 Z in this situation.
Theorem 2.1.15. [9] Let X be a normed space and Y and Z be subspaces of X.Then X = Y ⊕ Z if and only if for X ∩ Y = {0} and for every x in X, there arey ∈ Y and z ∈ Z such that x = y + z.
Definition 2.1.16. [9] A linear functional f is a linear operator from a normedspace X into the scalar filed K of X, where K is regarded as a normed space underthe usual norm on K.
If X is a normed space, then the set of all bounded linear functionals on X isdenoted by X∗. By Theorem 2.1.11, the normed space X∗ is immediately a Banachspace.
Theorem 2.1.17. [9] (Hahn-Banach extension theorem) Let X be a Banach spaceand Y a closed subspace of X. If f0 is a bounded linear functional on Y , then thereis a unique bounded linear functional f on X such that f(x) = f0(x) for all x ∈ Yand ‖f‖ = ‖f0‖.
Definition 2.1.18. [9] Let X be a normed space and Y a subspace of X. Theannihilator of Y , denoted by Y ⊥, is the set {f ∈ X∗ : f(x) = 0 for all x ∈ Y }.
Theorem 2.1.19. [9] If X is a normed space and Y is a subspace of X, then Y ⊥ isa closed subspace of X∗.
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Definition 2.1.20. [9] Let X and Y be Banach spaces. A linear operator T : X → Yis compact if T (B) is compact for all bounded subset B of X. The set of all compactoperators from X into Y will be denoted by K(X,Y ). For the case where X = Y ,we write K(X) instead of K(X, Y ).
Proposition 2.1.21. [9] Let X and Y be Banach spaces. Then the following hold.
(1) K(X, Y ) ⊆ B(X, Y ).
(2) K(X, Y ) is a closed subspace of B(X,Y ).
(3) If X = Y , then K(X) is an ideal of B(X).
Definition 2.1.22. [9] A linear operator T from a Banach space X into a Banachspace Y is said to be of finite rank if T (X) is finite dimensional.
Theorem 2.1.23. [9] A finite rank operator from a Banach space X into a Banachspace Y is bounded if and only if it is compact.
2.2 lp Spaces
Definition 2.2.1. [9] For 1 ≤ p ≤ ∞ and a sequence {λk}∞k=1 of complex numbers,the p-norm of {λk}∞k=1 is defined by
‖{λk}∞k=1‖p =
⎧⎪⎪⎪⎨⎪⎪⎪⎩( ∞∑
k=1
|λk|p)1/p
if 1 ≤ p < ∞,
sup{|λk| : k = 1, 2, 3, ...} if p = ∞.
For each 1 ≤ p < ∞, let
lp =
{{λk}∞k=1 ⊆ C :
∞∑k=1
|λk|p < ∞}
andl∞ = {{λk}∞k=1 ⊆ C : sup{|λk| : k = 1, 2, 3, ...} < ∞} .
Theorem 2.2.2. [9] (Holder’s inequality) For any 1 ≤ p ≤ ∞ with1
p+
1
q= 1 and
sequences x and y of complex numbers, ‖xy‖1 ≤ ‖x‖p ‖y‖q .
In particular, Holder’s inequality is also called Cauchy-Schwartz’s inequalitywhen p = q = 2. From Holder’s inequality, the following Minkowski’s inequality isobtained.
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Theorem 2.2.3. [9] (Minkowski’s inequality) For any 1 ≤ p ≤ ∞ and sequences xand y of complex numbers, ‖x + y‖p ≤ ‖x‖p + ‖y‖p .
Theorem 2.2.4. [9] For any 1 ≤ p ≤ ∞, the set lp endowed with the p-norm ‖·‖p isa Banach space.
For 1 ≤ p < ∞ with1
p+
1
q= 1, we define, for each x = {xk}∞k=1 ∈ lq, the
function fx : lp → C by
fx({yk}∞k=1) =∞∑
k=1
xkyk for all {yk}∞k=1 ∈ lp.
By Holder’s inequality, we have that the function fx is well-defined.
Theorem 2.2.5. [9] Let 1 ≤ p < ∞ with1
p+
1
q= 1. Then lq is isometrically
isomorphic to (lp)∗ by the isomorphism defined by x �→ fx.
The following result is closely related to the duality theorem stated above.
Theorem 2.2.6. [9] Let 1 ≤ p < ∞ with1
p+
1
q= 1. Then a sequence {xk}∞k=1 of
complex numbers belongs to lq if and only if {xkyk}∞k=1 belongs to l1 for all {yk}∞k=1
in lp.
2.3 Hilbert Spaces
Definition 2.3.1. [4] Let H be a vector space over a scalar K (K is either R or C),a semi-inner product on H is a function 〈·, ·〉 : H × H → K having the followingproperties:
(i) 〈αx + βy, z〉 = α〈x, z〉 + β〈y, z〉;(ii) 〈x, x〉 � 0;
(iii) 〈x, y〉 = 〈y, x〉.If 〈·, ·〉 has the following additional property:
(iv) if 〈x, x〉 = 0, then x = 0,
we call 〈·, ·〉 an inner product on H.
From (i), we have 〈0, y〉 = 〈0x, y〉 = 0〈x, y〉 = 0, and similarly, 〈x, 0〉 =0. In particular, 〈0, 0〉 = 0. Hence if 〈·, ·〉 is an inner-product, then 〈x, x〉 =0 if and only if x = 0. If 〈·, ·〉 is an inner-product on H, then
‖x‖ = 〈x, x〉1/2
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defines a norm on H. We call a vector space H equipped with an inner product on Han inner product space. Every inner product space is a normed space under the normdefined by ‖x‖ = 〈x, x〉1/2. If H equipped with the norm ‖ · ‖ is a Banach space, wecall H a Hilbert space.
Let, in the sequel, H and L be Hilbert spaces.
Definition 2.3.2. [4] If f, g ∈ H, then f and g are orthogonal if 〈f, g〉 = 0, insymbols, f ⊥ g. If A,B ⊆ H, we say that A and B are orthogonal and writeA ⊥ B provided f ⊥ g for every f in A and g in B. If A ⊆ H and f ∈ Hsatisfying {f} ⊥ A, then we write f ⊥ A. If A ⊆ H, then the set A⊥ is defined byA⊥ = {h ∈ H : h ⊥ g for all g ∈ A}.
Definition 2.3.3. [4] An orthonormal set in H is a subset E of H having the followingproperties:
(i) for e ∈ E , ‖e‖ = 1;
(ii) if e1, e2 ∈ E and e1 �= e2, then e1⊥e2.
An orthonormal basis for H is a maximal orthonormal set.
Proposition 2.3.4. [4] If E is an orthonormal set in H, then there is an orthonormalbasis for H that contains E .
Theorem 2.3.5. [4] If E is an orthonormal set in H and h ∈ H, then {e ∈ E :〈h, e〉 �= 0} is countable.
Theorem 2.3.6. [4] Let E be an orthonormal set in H. Then the following statementsare equivalent.
(1) E is an orthonormal basis.
(2) If h ∈ H and h ⊥ E, then h = 0.
(3)∨ E = H, where
∨ E is the smallest closed subspace of H containing E.
(4) h =∑{〈h, e〉e : e ∈ E} for all h ∈ H, where
∑{〈h, e〉e : e ∈ E} denotes the
limit of the net
{∑e∈F
〈h, e〉e : F is a finite subset of E}
.
Theorem 2.3.7. Any two orthonormal bases of H have the same cardinality.
Definition 2.3.8. [4] The dimension of H is the cardinality of an orthornomal basisand is denoted by dimH.
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Definition 2.3.9. [4] A subset D of H is said to be dense in H if D = H. H is saidto be separable if it has a countable subset which is dense in H.
Theorem 2.3.10. [4] Let H be an infinite dimensional Hilbert space. Then H isseparable if and only if dimH=ℵ0, where ℵ0 is the cardinality of the set of all positiveintegers.
Definition 2.3.11. [4] A function u : H× L → K is a sesquilinear form if for h, gin H, k, f in L, and α, β in K,
(i) u(αh + βg, k) = αu(h, k) + βu(g, k);
(ii) u(h, αk + βf) = αu(h, k) + βu(h, f).
Definition 2.3.12. [4] A sesquilinear form u is bounded if there is a constant Msuch that |u(h, k)| ≤ M ‖h‖ ‖k‖ for all h in H and k in L. The constant M is calleda bound for u.
Theorem 2.3.13. [4] If u : H×L → K is a bounded sesquilinear form with a boundM , then there are unique operators A in B(H,L) and B in B(L,H) such that
u(h, k) = 〈Ah, k〉 = 〈h,Bk〉
for all h in H and k in L and both ‖A‖ and ‖B‖ are not greater than M .
Definition 2.3.14. [4] If A ∈ B(H,L), then the unique operator B in B(L,H)satisfying 〈Ah, k〉 = 〈h,Bk〉 is called the adjoint of A and is denoted by A∗.
Proposition 2.3.15. [4] If A,B ∈ B(H), where B(H) = B(H,H) and α ∈ K, thenthe following hold.
(1) (αA + B)∗ = αA∗ + B∗.
(2) (AB)∗ = B∗A∗.
(3) A∗∗ = (A∗)∗ = A.
(4) If A is invertible in B(H), then (A∗)−1 = (A−1)∗.
Theorem 2.3.16. [4] Let H be an infinite dimensional separable Hilbert space withan orthonormal basis {en}. If T ∈ B(H) with the matrix representation A = [aji]with respect to the basis {en}, then the matrix representation of T ∗ with respect to{en} is the matrix [aji]
t.
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Definition 2.3.17. [4] A bounded linear operator A on a Hilbert space H is said tobe self-adjoint if A = A∗.
Theorem 2.3.18. [4] If T ∈ B(H,L), the following statements are equivalent.
(1) T is compact.
(2) T ∗ is compact.
(3) There is a sequence {Tn}of operators of finite rank such that ‖T − Tn‖ → 0.
Definition 2.3.19. [4] If A ∈ B(H), a scalar α is an eigenvalue of A if ker(A−αI) �={0}.
Definition 2.3.20. [4] If T ∈ B(H), then T is positive if 〈Th, h〉 ≥ 0 for all h ∈ H.
In symbols, this is denoted by T ≥ 0. Note that every positive operator on acomplex Hilbert space is self-adjoint.
Theorem 2.3.21. [4] If T is a positive compact operator on a Hilbert space H, thenthere is a unique positive compact operator A such that A2 = T .
Definition 2.3.22. [4] If T is a positive compact operator on a Hilbert space H, thenthe unique positive compact operator A such that A2 = T according to Theorem2.3.21 is called the positive square root of T and denoted by |T |.
Definition 2.3.23. [4] A partial isometry is a linear operator W such that ‖Wh‖ =‖h‖ for all h ∈ (ker W )⊥. The space (ker W )⊥ is called the initial space of W and thespace ran W is called the final space of W .
Theorem 2.3.24. [4] (Polar Decomposition) If T ∈ B(H), then there is a partialisometry W with (ker T )⊥ as its initial space and ran T as its final space such thatT = W |T |. Moreover, if T = UP where P ≥ 0 and U is a partial isometry withker U = ker P , then P = |T | and U = W .
Theorem 2.3.25. [4] (Spectral Theorem) If T is a compact self-adjoint operator onH, then T has only a countable number of distinct eigenvalues. If {λ1, λ2, ...} are thedistinct nonzero eigenvalue of T with |λ1| ≥ |λ2| ≥ |λ3| ≥ ..., and Pn is the projectionof H onto ker(T − λn), then PnPm = PmPn = 0 if n �= m, each λn is an real, and
T =∞∑
n=1
λnPn,
where the series converges to T in the metric defined by the norm of B(H).
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Corollary 2.3.26. [4] With the notation of Spectral Theorem. One has the following.
(1) ker T = span(⋃∞
n=1 PnH) = (ran T )⊥;
(2) each Pn has finite rank;
(3) ‖T‖ = sup{|λn| : n ≥ 1} and λn → 0 as n → ∞.
Let T be a compact self-adjoint operator on H. By Spectral Theorem, Thas precisely finite or countable number of distinct eigenvalues. Let {λn}∞n=1 be thesequence of eigenvalues of T with |λ1| ≥ |λ2| ≥ |λ3| ≥ .... For each n, let Nn be thedimension of ker(T − λn), and let {μn}∞n=1 = {λ1, ..., λ1︸ ︷︷ ︸
N1
, λ2, ..., λ2︸ ︷︷ ︸N2
, ...}. If T has only
k eigenvalues, then we let μn = 0 for all for n > N1 + N2 + ... + Nk.
Corollary 2.3.27. [4] If T is a compact self-adjoint operator on H, then there is anorthonormal basis {en} for (ker T )⊥ such that
Th =∞∑
n=1
μn〈h, en〉en
for all h ∈ H.
2.4 Schatten p-Classes
Let H be an infinite dimensional separable Hilbert space and K a compactoperator on H. Since 0 ≤ ‖Kh‖2 = 〈Kh,Kh〉 = 〈K∗Kh, h〉 for all h ∈ H, itfollows that K∗K is a positive compact operator. Whence, by Theorem 2.3.21,there is a unique positive compact operator |K| such that |K|2 = K∗K. Since |K| is
positive, |K| is self-adjoint. Thus, by Corollary 2.3.27, we have |K|h =∞∑
n=1
μn〈h, en〉en
for all h ∈ H, where {μn}∞n=1 is the sequence of eigenvalues of |K| and {en}∞n=1 isan orthonormal basis for (ker |K|)⊥. Notice that ker K = ker |K| due to the factthat ‖Kh‖2 = 〈Kh, Kh〉 = 〈h,K∗Kh〉 = 〈h, |K|2h〉 = 〈|K|h, |K|h〉 = ‖|K|h‖2 forall h ∈ H. We call the sequence {μn}∞n=1 the sequence of singular values of thecompact operator K and denote μn by sn(K) for all n. By Corollary 2.3.26, we have
‖K‖ = s1(K) ≥ s2(K) ≥ ... ≥ 0 and limn→∞
sn(K) = 0.
For 1 ≤ p ≤ ∞, let
Cp = {K ∈ K(H) : {sk(K)}∞k=1 ∈ lp}.The set Cp is called the Schatten p-class. We define, for 1 ≤ p < ∞, the norm ‖·‖p
on Cp by
‖K‖p =
( ∞∑n=1
sn(K)p
)1/p
.
For p = ∞, we define ‖K‖∞ = supn
sn(K). It is obvious that for any compact
operator K on H, ‖K‖∞ = s1(K) = ‖K‖. Thus C∞ = K(H).
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Theorem 2.4.1. [5] If K ∈ C1, then for each othonormal basis {en} of H the sum∞∑
n=1
〈Ken, en〉 is absolutely convergent and
∞∑n=1
〈Ken, en〉 =∞∑
n=1
sn(K)〈Uen, en〉,
where U is the unique partial isometry such that K = U |K|.
Definition 2.4.2. [5] For each K ∈ C1, the number
∞∑n=1
〈Ken, en〉,
where {en} is an othonormal basis of H, is called the trace of K.
Remark 2.4.3. [5] If {en} is an ordered othonormal basis of H and K ∈ C1 withthe matrix representation A with respect to {en}, then the trace of K is exactly thesum of all entries in the main diagonal of A.
Theorem 2.4.4. [5] For each 1 < p ≤ ∞ with1
p+
1
q= 1, (Cp)∗ ∼= Cq.
Theorem 2.4.5. [5] (C1)∗ ∼= B(H).
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Chapter 3
Duality of Sequence Spaces of Infinite
Matrices
3.1 Basic Results
Recall that, for any 1 ≤ r < ∞, the set L r of sequences of infinite matricesis defined by
L r =
{{[a
(k)ji
]}∞
k=1⊆ M∞ :
[ ∞∑k=1
∣∣∣a(k)ji
∣∣∣r] ∈ B(l2)
}.
It is clear that if {Ak}∞k=1 ∈ L r, then Ak is necessarily a member of the absoluteSchur algebra Sr
2,2(C) for all k.
The following theorem was first stated in [3] by A. Charearnpol. It is a gen-eralization of the characterization of the sequence spaces Ob provided by J. Rakbudand S.-C. Ong in [11].
Theorem 3.1.1. Let{[
a(k)ji
]}∞
k=1be a sequence in B(l2) with a
(k)ji ≥ 0 for all i, j, k.
(1) The sequence
{n∑
k=1
[a
(k)ji
]}∞
n=1
is bounded in B(l2) if and only if
[ ∞∑k=1
a(k)ji
]∈
B(l2).
(2) If
[ ∞∑k=1
a(k)ji
]∈ B(l2), then
∥∥∥∥∥[ ∞∑
k=1
a(k)ji
]∥∥∥∥∥ = supn
∥∥∥∥∥n∑
k=1
[a
(k)ji
]∥∥∥∥∥.
From the above theorem, the following characterization of the set L r is im-mediately obtained.
Corollary 3.1.2. Let {Ak}∞k=1 be a sequence in M∞ and 1 ≤ r < ∞. Then thefollowing are equivalent:
(1) {Ak}∞k=1 belongs to L r;
14
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(2) Ak ∈ Sr2,2(C) for all k and the sequence
{n∑
k=1
A[r]k
}∞
n=1
is bounded in B(l2);
(3) the sequence
{∥∥∥∥∥n∑
k=1
A[r]k
∥∥∥∥∥}∞
n=1
is bounded.
For any sequence{[
a(k)ji
]}∞
k=1in M∞ and 1 ≤ r < ∞, we define
∥∥∥∣∣∣{[a(k)ji
]}∞
k=1
∣∣∣∥∥∥r
=
⎧⎪⎪⎨⎪⎪⎩∥∥∥∥∥[ ∞∑
k=1
∣∣∣a(k)ji
∣∣∣r]∥∥∥∥∥1/r
if{[
a(k)ji
]}∞
k=1∈ L r,
∞ otherwise.
The following Holder-type inequality was first established in [2] by Chaisuriyaand Ong. It is useful for the research.
Theorem 3.1.3. (Holder-type inequality) For any A,B ∈ M∞ and 1 < r < ∞ with1
r+
1
r∗= 1, ∥∥(A • B)[1]
∥∥ ≤ ∥∥A[r]∥∥1/r ∥∥B[r∗]
∥∥1/r∗
under the conventions that ∞· 0 = 0 ·∞ = 0, ∞·α = α ·∞ = ∞ for all positive realnumber α and ∞ ·∞ = ∞.
The Holder and Minkowski-type inequalities below are extensions of the onesin [11].
Theorem 3.1.4. (Holder-type inequality for sequences of matrices) For any se-quences {Ak}∞k=1 and {Bk}∞k=1 in M∞,
‖|{Ak • Bk}∞k=1|‖1 ≤ ‖|{Ak}∞k=1|‖r ‖|{Bk}∞k=1|‖r∗ ,
where 1 < r < ∞ with1
r+
1
r∗= 1, under the same convention as in Theorem 3.1.3.
Proof. Let{
Ak =[a
(k)ji
]}∞
k=1and
{Bk =
[b(k)ji
]}∞
k=1be sequences in M∞. If either
‖|{Ak}∞k=1|‖r or ‖|{Bk}∞k=1|‖r∗ is ∞, then we are done. Suppose that both ‖|{Ak}∞k=1|‖r
and ‖|{Bk}∞k=1|‖r∗ are finite. Then
[ ∞∑k=1
∣∣∣a(k)ji
∣∣∣r] and
[ ∞∑k=1
∣∣∣b(k)ji
∣∣∣r∗] belong to ∈ B(l2).
Thus, by Holder’s inequality, we have for each i, j that
∞∑k=1
∣∣∣a(k)ji b
(k)ji
∣∣∣ ≤ ( ∞∑k=1
∣∣∣a(k)ji
∣∣∣r)1/r ( ∞∑k=1
∣∣∣b(k)ji
∣∣∣r∗)1/r∗
< ∞.
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Hence the matrix
[ ∞∑k=1
∣∣∣a(k)ji b
(k)ji
∣∣∣] ∈ M∞. We want to show that
[ ∞∑k=1
∣∣∣a(k)ji b
(k)ji
∣∣∣] ∈B(l2) and ‖|{Ak • Bk}∞k=1|‖1 ≤ ‖|{Ak}∞k=1|‖r ‖|{Bk}∞k=1|‖r∗ . By the Holder-type inequal-ity, we have∥∥∥∥∥
[ ∞∑k=1
∣∣∣a(k)ji b
(k)ji
∣∣∣]∥∥∥∥∥ ≤∥∥∥∥∥∥⎡⎣( ∞∑
k=1
∣∣∣a(k)ji
∣∣∣r)1/r ( ∞∑k=1
∣∣∣b(k)ji
∣∣∣r∗)1/r∗⎤⎦∥∥∥∥∥∥
=
∥∥∥∥∥∥⎡⎣( ∞∑
k=1
∣∣∣a(k)ji
∣∣∣r)1/r⎤⎦ •
⎡⎣( ∞∑k=1
∣∣∣b(k)ji
∣∣∣r∗)1/r∗⎤⎦∥∥∥∥∥∥
≤∥∥∥∥∥[ ∞∑
k=1
∣∣∣a(k)ji
∣∣∣r]∥∥∥∥∥1/r ∥∥∥∥∥
[ ∞∑k=1
∣∣∣b(k)ji
∣∣∣r∗]∥∥∥∥∥1/r∗
= ‖|{Ak}∞k=1|‖r ‖|{Bk}∞k=1|‖r∗ < ∞.
This implies that
[ ∞∑k=1
∣∣∣a(k)ji b
(k)ji
∣∣∣] ∈ B(l2), which is equivalent to that {Ak •Bk}∞k=1 ∈L 1, and ‖|{Ak • Bk}∞k=1|‖1 ≤ ‖|{Ak}∞k=1|‖r ‖|{Bk}∞k=1|‖r∗ .
Theorem 3.1.5. (Minkowski-type inequality for sequences of matrices) For any se-quences {Ak}∞k=1 and {Bk}∞k=1 in M∞ and 1 ≤ r < ∞,
‖|{Ak + Bk}∞k=1|‖r ≤ ‖|{Ak}∞k=1|‖r + ‖|{Bk}∞k=1|‖r
under the conventions that ∞ + α = α + ∞ = ∞ for all non-negative real number αand ∞ + ∞ = ∞.
Proof. For the case where either ‖|{Ak}∞k=1|‖r = ∞ or ‖|{Bk}∞k=1|‖r = ∞, there isnothing to prove. Suppose that both ‖|{Ak}∞k=1|‖r and ‖|{Bk}∞k=1|‖r are finite. Weassume first that 1 < r < ∞. Then by the Holder-type inequality for sequences ofmatrices, we have for each positive integer n that∥∥∥∥∥
n∑k=1
(Ak + Bk)[r]
∥∥∥∥∥ =
∥∥∥∥∥n∑
k=1
(Ak + Bk)[1] • (Ak + Bk)
[r−1]
∥∥∥∥∥≤
∥∥∥∥∥n∑
k=1
(A
[1]k + B
[1]k
)• (Ak + Bk)
[r−1]
∥∥∥∥∥=
∥∥∥∥∥n∑
k=1
A[1]k • (Ak + Bk)
[r−1] +n∑
k=1
B[1]k • (Ak + Bk)
[r−1]
∥∥∥∥∥≤
∥∥∥∥∥n∑
k=1
A[1]k • (Ak + Bk)
[r−1]
∥∥∥∥∥ +
∥∥∥∥∥n∑
k=1
B[1]k • (Ak + Bk)
[r−1]
∥∥∥∥∥
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≤∥∥∥∥∥
n∑k=1
A[r]k
∥∥∥∥∥1/r ∥∥∥∥∥
n∑k=1
((Ak + Bk)
[r−1])[r∗]
∥∥∥∥∥1/r∗
+
∥∥∥∥∥n∑
k=1
B[r]k
∥∥∥∥∥1/r ∥∥∥∥∥
n∑k=1
((Ak + Bk)
[r−1])[r∗]
∥∥∥∥∥1/r∗
=
⎛⎝∥∥∥∥∥n∑
k=1
A[r]k
∥∥∥∥∥1/r
+
∥∥∥∥∥n∑
k=1
B[r]k
∥∥∥∥∥1/r
⎞⎠∥∥∥∥∥n∑
k=1
(Ak + Bk)[r]
∥∥∥∥∥1/r∗
,
where1
r+
1
r∗= 1, which implies that
∥∥∥∥∥n∑
k=1
(Ak + Bk)[r]
∥∥∥∥∥1/r
≤∥∥∥∥∥
n∑k=1
A[r]k
∥∥∥∥∥1/r
+
∥∥∥∥∥n∑
k=1
B[r]k
∥∥∥∥∥1/r
≤ ‖|{Ak}∞k=1|‖r + ‖|{Bk}∞k=1|‖r .
For the case where r = 1, we easily have for each positive integer n that∥∥∥∥∥n∑
k=1
(Ak + Bk)[1]
∥∥∥∥∥ ≤∥∥∥∥∥
n∑k=1
A[1]k +
n∑k=1
B[1]k
∥∥∥∥∥≤
∥∥∥∥∥n∑
k=1
A[1]k
∥∥∥∥∥ +
∥∥∥∥∥n∑
k=1
B[1]k
∥∥∥∥∥≤ ‖|{Ak}∞k=1|‖1 + ‖|{Bk}∞k=1|‖1
as well. Thus, by Corollary 3.1.2, the sequence {Ak + Bk}∞k=1 belongs to L r and
‖|{Ak + Bk}∞k=1|‖r ≤ ‖|{Ak}∞k=1|‖r + ‖|{Bk}∞k=1|‖r
for all 1 ≤ r < ∞. The proof is complete.
The following lemma was first stated and proved in [10]. It is a beautifulconsequence of the Holder-type inequality.
Lemma 3.1.6. For any 1 ≤ r < ∞ and matrices A and B in Sr2,2(C),∥∥A[r] − B[r]
∥∥ ≤ (‖|A|‖2,2,r + ‖|B|‖2,2,r) ‖|A − B|‖2,2,r .
The proposition below was first stated and proved in [10] as well. We can seethat it follows easily from the lemma above.
Proposition 3.1.7. For any 1 ≤ r < ∞, the map A �→ A[r] from Sr2,2(C) into B(l2)
is continuous.
Theorem 3.1.8. For each 1 ≤ r < ∞, the set L r equipped with the norm ‖|·|‖r is aBanach space.
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Proof. From Minkowski’s inequality for sequences of matrices, we have that the setL r endowed with the norm ‖|·|‖r is a normed space. To see that it is a Banach space,
let{An =
{A
(n)k
}∞
k=1
}∞
n=1be a Cauchy sequence in L r. As we have for each k that∥∥∥∣∣∣A(n)
k − A(m)k
∣∣∣∥∥∥2,2,r
≤ ‖|An − Am|‖r for all n,m,
it follows that the sequence{
A(n)k
}∞
n=1is a Cauchy sequence in Sr
2,2(C) for all k. So,
for each k, we obtain by the completeness of Sr2,2(C) that there exists an Ak in Sr
2,2(C)
such that A(n)k → Ak. Let A = {Ak}∞k=1. We claim that A ∈ L r and An → A. To
prove these, let ε > 0 be given. Then there is a positive integer N such that for eachpositive integer K,∥∥∥∥∥
K∑k=1
(A
(n)k − A
(m)k
)[r]
∥∥∥∥∥1/r
≤ ‖|An − Am|‖r <ε
2for all n,m ≥ N. (∗)
Since A(m)k → Ak in Sr
2,2(C) for all k, it follows for each fixed n that A(n)k − A
(m)k →
A(n)k −Ak in Sr
2,2(C) for all k. Thus, by Proposition 3.1.7, we obtain for each fixed n
that(A
(n)k − A
(m)k
)[r]
→(A
(n)k − Ak
)[r]
in B(l2) for all k. From this we have for each
fixed n and K thatK∑
k=1
(A
(n)k − A
(m)k
)[r]
→K∑
k=1
(A
(n)k − Ak
)[r]
in B(l2). Whence, by
taking the limits as m → ∞ on both sides of (∗), we obtain by the continuity of theoperator norm on B(l2) that for each n ≥ N ,∥∥∥∥∥
K∑k=1
(A
(n)k − Ak
)[r]
∥∥∥∥∥1/r
≤ ε
2for all K ≥ 1.
Therefore, by Theorem 3.1.1,
‖|An − A|‖r = supK
∥∥∥∥∥K∑
k=1
(A
(n)k − Ak
)[r]
∥∥∥∥∥1/r
< ε for all n ≥ N. (∗∗)
The inequality (∗∗) yields that AN −A belongs to L r, which implies that A = AN −(AN − A) is an element of L r. Consequently, by (∗∗) again, we get An → A.
3.2 Duality
In this section, we study the duality of the sequence spaces L r. The aimis to decompose the dual space (L r)∗ of L r as an l1 direct-sum of its two closedsubspaces. Before getting the results, we need some notational conventions.
For any z ∈ C, we define the function sgn(·) on C by
sgn(z) =
⎧⎪⎨⎪⎩z
|z| if z �= 0,
1 if z = 0.
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For any sequences A = {Ak}∞k=1 and B = {Bk}∞k=1 in M∞ and any positive integern, we let A • B = {Ak • Bk}∞k=1, An� = {(Ak)n�}∞k=1, An� = {(Ak)n�}∞k=1, andAn] = {A1, A2, ..., An, 0, 0, ...}. It is clear that
(AK]
)n�
= (An�)K] for all positive
integers n and K. Notice that for each 1 ≤ r < ∞, if A ={
Ak =[a
(k)ji
]}∞
k=1∈ L r,
then each of the following holds true:
(i) ‖|An� |‖r =
∥∥∥∥∥[ ∞∑
k=1
∣∣∣a(k)ji
∣∣∣r]n�
∥∥∥∥∥1/r
,
(ii)∥∥∣∣AK]
∣∣∥∥r
=
∥∥∥∥∥K∑
k=1
A[r]k
∥∥∥∥∥1/r
=
∥∥∥∥∥[
K∑k=1
∣∣∣a(k)ji
∣∣∣r]∥∥∥∥∥1/r
and
(iii) ‖|An� − A|‖r = ‖|{(Ak)n� − Ak}∞k=1|‖r =
∥∥∥∥∥[ ∞∑
k=1
∣∣∣a(k)ji
∣∣∣r]n�
−[ ∞∑
k=1
∣∣∣a(k)ji
∣∣∣r]∥∥∥∥∥1/r
,
for all n and K. The first and second equations imply that ‖|A|‖r = supn
‖|An� |‖r
and ‖|A|‖r = supK
∥∥∣∣AK]
∣∣∥∥r
respectively. And the last one implies that the ma-
trix
[ ∞∑k=1
∣∣∣a(k)ji
∣∣∣r] is compact if and only if ‖|An� − A|‖r → 0. For each A = [aji]
in M∞ and positive integer k, let∑
A =∞∑
j=1
∞∑i=1
aji if the series converges, let
sgnA=[sgn(aji)], and let s(A; k) be the sequence whose k-th term is the matrix Aand all other terms are 0. Finally, for any λ ∈ C and pair (j, i) of positive integers, letE(λ; (j, i)) be the matrix whose (j, i)-th entry is the number λ and all other entriesare 0.
On the classical sequence spaces lp, there is a result closely related to their
duality as follows: for 1 ≤ p < ∞ with1
p+
1
q= 1, a sequence x belongs to lq if and
only if x “Schur multiplies”every y in lp into l1. An analogue of this result is alsoobtained for our sequence spaces L r of infinite matrices.
Theorem 3.2.1. Let 1 < r < ∞ with1
r+
1
r∗= 1.
(1) {Ak}∞k=1 ∈ L r∗ if and only if {Ak • Bk}∞k=1 ∈ L 1 for all {Bk}∞k=1 ∈ L r.
(2) If {Ak}∞k=1 ∈ L r∗, then
‖|{Ak}∞k=1|‖r∗ = sup{‖|{Ak • Bk}∞k=1|‖1 : {Bk}∞k=1 ∈ L r, ‖|{Bk}∞k=1|‖r ≤ 1}.
Proof. (1). Let A ={
Ak =[a
(k)ji
]}∞
k=1be a sequence in M∞. Suppose that {Ak •
Bk}∞k=1 ∈ L 1 for all {Bk}∞k=1 ∈ L r. We want to show that {Ak}∞k=1 ∈ L r∗ . Bythe assumption, a map Ψ : L r → L 1 can be defined as follows: Ψ({Bk}∞k=1) ={Ak •Bk}∞k=1 for all {Bk}∞k=1 ∈ L r. For any positive integer n, let Ψn : L r → L 1 be
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20
defined by Ψn({Bk}∞k=1) = An] • B for all B = {Bk}∞k=1 ∈ L r. Then by the Holder-type inequality for sequences of matrices, we have for every B = {Bk}∞k=1 ∈ L r
that‖|Ψn({Bk}∞k=1)|‖1 =
∥∥∣∣An] • B∣∣∥∥
1≤ ∥∥∣∣An]
∣∣∥∥r∗ ‖|B|‖r .
So the operator Ψn is bounded for all n. For each{
Bk =[b(k)ji
]}∞
k=1∈ L r, we have
‖|Ψn({Bk}∞k=1)|‖1 =
∥∥∥∥∥[
n∑k=1
∣∣∣a(k)ji b
(k)ji
∣∣∣]∥∥∥∥∥ ≤∥∥∥∥∥[ ∞∑
k=1
∣∣∣a(k)ji b
(k)ji
∣∣∣]∥∥∥∥∥= ‖|{Ak • Bk}∞k=1|‖1 for all n.
Hence, by the uniform boundedness principle, the set {‖Ψn‖ : n = 1, 2, 3, ...} isbounded. For every B = {Bk}∞k=1 ∈ L r with ‖|B|‖r ≤ 1, we have by Theorem 3.1.1that
‖|Ψ(B)|‖1 = supn
∥∥∥∥∥n∑
k=1
(Ak • Bk)[1]
∥∥∥∥∥ = supn
∥∥∣∣An] • B∣∣∥∥
1
= supn
‖|Ψn(B)|‖1 ≤ supn
‖Ψn‖ .
Thus, by the boundedness of the set {‖Ψn‖ : n = 1, 2, 3, ...}, the operator Ψ is
bounded. Next, let D ={
A[r∗−1]k
}∞
k=1. Then (Dn�)K] ∈ L r for all n,K. Thus∥∥∥∥∥
(K∑
k=1
A[r∗]k
)n�
∥∥∥∥∥ =
∥∥∥∥∥K∑
k=1
(A
[r∗]k
)n�
∥∥∥∥∥ =
∥∥∥∥∥K∑
k=1
A[1]k •
(A
[r∗−1]k
)n�
∥∥∥∥∥=
∥∥∥∣∣∣A • (Dn�)K]
∣∣∣∥∥∥1
=∥∥∥∣∣∣Ψ(
(Dn�)K]
)∣∣∣∥∥∥1
≤ ‖Ψ‖∥∥∥∣∣∣(Dn�)K]
∣∣∣∥∥∥r
= ‖Ψ‖∥∥∥∥∥(
K∑k=1
(A
[r∗−1]k
)[r])
n�
∥∥∥∥∥1/r
= ‖Ψ‖∥∥∥∥∥(
K∑k=1
A[r∗]k
)n�
∥∥∥∥∥1/r
for all n, K. ()
It follows that ∥∥∥∥∥(
K∑k=1
A[r∗]k
)n�
∥∥∥∥∥ ≤ ‖Ψ‖r∗ for all n,K.
Whence, by Theorem 1.1(2), we obtain for each K thatK∑
k=1
A[r∗]k ∈ B(l2) and by
Theorem 1.1(3), ∥∥∥∥∥K∑
k=1
A[r∗]k
∥∥∥∥∥ = supn
∥∥∥∥∥(
K∑k=1
A[r∗]k
)n�
∥∥∥∥∥ ≤ ‖Ψ‖r∗ .
Therefore, by Corollary 3.1.2, the sequence A belongs to L r∗ . Conversely, supposethat A ∈ L r∗ . Then for any {Bk}∞k=1 ∈ L r, we have by the Holder-type inequalityfor sequences of matrices that {Ak • Bk}∞k=1 ∈ L 1.
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21
(2). Suppose that {Ak}∞k=1 ∈ L r∗ . Then by (1), the linear operator Ψ :L r → L 1 defined by {Bk}∞k=1 �→ {Ak • Bk}∞k=1 is well-defined, and by the Holder-type inequality for sequences of matrices, it is obvious that ‖Ψ‖ ≤ ‖|{Ak}∞k=1|‖r∗ .By the same argument as given in the proof of (1) (see the argument to obtain theinequality ()), we have ∥∥∥∥∥
n∑k=1
A[r∗]k
∥∥∥∥∥1/r∗
≤ ‖Ψ‖ for all n.
It follows from Theorem 3.1.1 that ‖|{Ak}∞k=1|‖r∗ ≤ ‖Ψ‖. Consequently, we obtain
‖|{Ak}∞k=1|‖r∗ = ‖Ψ‖ = sup{‖|{Ak • Bk}∞k=1|‖1 : {Bk}∞k=1 ∈ L r, ‖|{Bk}∞k=1|‖r ≤ 1}as required. The proof is complete.
For each 1 ≤ r < ∞, let
L rκ =
{{[a
(k)ji
]}∞
k=1⊆ M∞ :
[ ∞∑k=1
∣∣∣a(k)ji
∣∣∣r] ∈ K(l2)
}.
The following results on the sets L rκ are evident.
(i) L rκ � L r.
(ii) A sequence A in M∞ belongs to L rκ if and only if ‖|A − An� |‖r → 0.
(iii) If a sequence A belongs to L rκ , then A − An� belongs to L r
κ for all n.
The following theorem is a more general version of the characterization of thesequence space Oκ provided by Rakbud et al. in [11].
Theorem 3.2.2. Let{
Ak =[a
(k)ji
]}∞
k=1be a sequence in M∞ with a
(k)jk ≥ 0 for all
i, j, k. Then
[ ∞∑k=1
a(k)ji
]∈ K(l2) if and only if Ak ∈ K(l2) for all k and the sequence{
n∑k=1
Ak
}∞
k=1
converges in B(l2).
Proof. Suppose that the matrix A =
[ ∞∑k=1
a(k)ji
]is compact. Then for each k, we have
by Theorem 1.1(1) that Ak ∈ B(l2) and
‖Ak − (Ak)n�‖ ≤ ‖A − An�‖ → 0.
Thus Ak is compact for all k. To see that the sequence
{n∑
k=1
Ak
}∞
k=1
converges in
B(l2), let ε > 0 be given. Then by the compactness of the matrix A, there exists
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22
a positive integer N such that ‖AN� − A‖ <ε
3. As the series
∞∑k=1
a(k)ji converges for
all 1 ≤ j, i ≤ N , there is a positive integer K0 such that for each 1 ≤ j, i ≤ N ,∞∑
k=K
a(k)ji <
ε
3N3/2for all K ≥ K0. Hence for each K ≥ K0,
∥∥∥∥∥K∑
k=1
Ak − A
∥∥∥∥∥ ≤∥∥∥∥∥AN� −
(K∑
k=1
Ak
)N�
∥∥∥∥∥ +
∥∥∥∥∥(
K∑k=1
Ak
)N�
−K∑
k=1
Ak
∥∥∥∥∥+ ‖AN� − A‖
≤⎧⎨⎩
N∑j=1
(N∑
i=1
∞∑k=K
a(k)ji
)2⎫⎬⎭
1/2
+ 2 ‖AN� − A‖
<ε
3+
2ε
3= ε.
This yields∞∑
k=1
Ak = A in B(l2). Conversely, suppose that Ak is compact for all k and
that the sequence
{n∑
k=1
Ak
}∞
n=1
converges in B(l2). It is clear that∞∑
k=1
Ak =
[ ∞∑k=1
a(k)ji
].
Since K(l2) is closed in B(l2), it follows that∞∑
k=1
Ak is compact. Thus we obtain that[ ∞∑k=1
a(k)ji
]is compact as required.
The following characterization of the set L rκ is an immediate consequence of
Theorem 3.2.2 above.
Corollary 3.2.3. Let {Ak}∞k=1 be a sequence in M∞ and 1 ≤ r < ∞. Then
{Ak}∞k=1 ∈ L rκ if and only if A
[r]k is compact for all k and the sequence
{n∑
k=1
A[r]k
}∞
k=1
converges in B(l2).
Theorem 3.2.4. For each 1 ≤ r < ∞, the set L rκ is a Banach subspace of L r.
Proof. For any matrix A ∈ M∞ and positive integer n, we let here for convenienceA�n = A − An−1� . We will show first that L r
κ is a normed subspace of L r. Let{Ak}∞k=1, {Bk}∞k=1 ∈ L r
κ . Then
‖|{(Ak + Bk)�n}∞k=1|‖r = ‖|{(Ak)�n}∞k=1 + {(Bk)�n}∞k=1|‖r
≤ ‖|{(Ak)�n}∞k=1|‖r+ ‖|{(Bk)�n}∞k=1|‖r
→ 0.
Thus L rκ is closed under addition. It clear that λ{Ak}∞k=1 ∈ L r
κ for any complexnumber λ. Hence L r
κ is a normed subspace of L r. To show that L rκ is a Ba-
nach space, it suffices to show that L rκ is a closed subspace of L r. Suppose that
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23
{An =
{A
(n)k
}∞
k=1
}∞
n=1is a sequence in L r
κ converging to an element A = {Ak}∞k=1
in L r, and let ε > 0 be given. Then there is a positive integer N such that
‖|AN − A|‖r <ε
2.
Due to the membership of AN in L rκ , we have that there exists a positive integer J0
such that ∥∥∥∥∣∣∣∣{(A
(N)k
)�J
}∞
k=1
∣∣∣∣∥∥∥∥r
<ε
2for all J ≥ J0.
It follows that
‖|{(Ak)�J}∞k=1|‖r
≤∥∥∥∥∣∣∣∣{(
A(N)k
)�J
}∞
k=1
− {(Ak)�J}∞k=1
∣∣∣∣∥∥∥∥r
+
∥∥∥∥∣∣∣∣{(A
(N)k
)�J
}∞
k=1
∣∣∣∣∥∥∥∥r
=
∥∥∥∥∣∣∣∣{(A
(N)k − Ak
)�J
}∞
k=1
∣∣∣∣∥∥∥∥r
+
∥∥∥∥∣∣∣∣{(A
(N)k
)�J
}∞
k=1
∣∣∣∣∥∥∥∥r
≤ ‖|AN − A|‖r +
∥∥∥∥∣∣∣∣{(A
(N)k
)�J
}∞
k=1
∣∣∣∣∥∥∥∥r
<ε
2+
ε
2= ε for all J ≥ J0.
Consequently, {Ak}∞k=1 belongs to L rκ .
The following is the main theorem in this thesis. It tells us that the annihilator(L r
κ )⊥ of L rκ is complemented in (L r)∗. Furthermore, the norm of the decomposition
of any bounded linear functional on L r is additive.
Theorem 3.2.5. Let 1 ≤ r < ∞. Then the following hold.
(1) The annihilator (L rκ )⊥ of L r
κ is a non-trivial closed subspace of the dual (L r)∗
of L r.
(2) There is a subspace P of (L r)∗ such that P is isometrically isomorphic to(L r
κ )∗ and (L r)∗ = P ⊕ (L rκ )⊥ .
(3) For any f ∈ (L r)∗, the decomposition f = g+h, where g ∈ P and h ∈ (L rκ )⊥,
satisfies ‖f‖ = ‖g‖ + ‖h‖.Proof. (1). By the Hahn-Banach extension theorem, we have in general that if A isa non-trivial closed subspace of a Banach space X, then the annihilator A⊥ of A isa non-trivial closed subspace of the dual X∗ of X. Thus, by this fact, the assertion(1) holds.
(2). Let ϕ ∈ (L r)∗. For each k, let ϕk : Sr2,2(C) → C be defined by
ϕk(A) = ϕ(s(A; k)) for all A ∈ Sr2,2(C). It is easy to see that ϕk is linear and
‖ϕk‖ ≤ ‖ϕ‖ for all k. Hence, for each k, the map ϕk belongs to (Sr2,2(C))∗. Next,
let B(ϕ)k = [ϕk(E(1; (j, i)))] for all k, and let B(ϕ) =
{B
(ϕ)k
}∞
k=1. We want to
show first that∞∑
k=1
∑(Ak • B
(ϕ)k
)[1]
< ∞ for all {Ak}∞k=1 ∈ L r. To see this,
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24
let A ={
Ak =[a
(k)ji
]}∞
k=1∈ L r. Notice that A • B(ϕ) belongs to L r due to
the fact that∞∑
k=1
∣∣∣a(k)ji ϕk(E(1; (j, i)))
∣∣∣r ≤ ‖ϕ‖r∞∑
k=1
∣∣∣a(k)ji
∣∣∣r for all j, i. For each k, let
Ak =[(
sgn(ϕk
(E
(a
(k)ji ; (j, i)
))))a
(k)ji
], and let A =
{Ak
}∞
k=1. Then A ∈ L r with
the same norm as A. Let ν, μ and K be positive integers, and let n = max{ν, μ}.Then
K∑k=1
ν∑j=1
μ∑i=1
∣∣∣a(k)ji ϕk(E(1; (j, i)))
∣∣∣ <
K∑k=1
n∑j=1
n∑i=1
∣∣∣ϕk
(E
(a
(k)ji ; (j, i)
))∣∣∣=
K∑k=1
n∑j=1
n∑i=1
(sgn
(ϕk
(E
(a
(k)ji ; (j, i)
))))ϕk
(E
(a
(k)ji ; (j, i)
))
=K∑
k=1
n∑j=1
n∑i=1
ϕk
(E
((sgn
(ϕk
(E
(a
(k)ji ; (j, i)
))))a
(k)ji ; (j, i)
))
=K∑
k=1
ϕk
(n∑
j=1
n∑i=1
E((
sgn(ϕk
(E
(a
(k)ji ; (j, i)
))))a
(k)ji ; (j, i)
))
=K∑
k=1
ϕk
((Ak
)n�
)=
K∑k=1
ϕ
(s
((Ak
)n�
; k
))
= ϕ
(K∑
k=1
s
((Ak
)n�
; k
))= ϕ
((An�
)K]
)≤ ‖ϕ‖
∥∥∥∥∣∣∣∣(An�
)K]
∣∣∣∣∥∥∥∥r
≤ ‖ϕ‖∥∥∥∣∣∣A∣∣∣∥∥∥
r= ‖ϕ‖ ‖|A|‖r .
It follows that ∞∑k=1
∞∑j=1
∞∑i=1
∣∣∣a(k)ji ϕk(E(1; (j, i)))
∣∣∣ ≤ ‖ϕ‖ ‖|A|‖r .
From this result, we can define a bounded linear functional ψϕ on L r by {Ak}∞k=1 �→∞∑k=1
∑Ak • B
(ϕ)k with ‖ψϕ‖ ≤ ‖ϕ‖. Notice that for any A = {Ak}∞k=1 ∈ L r and
positive integer K, we have by the absolute convergence of the series∑
Ak • B(ϕ)k
(k = 1, 2, ..., K) that
ψϕ
(AK]
)=
K∑k=1
∑Ak • B
(ϕ)k =
K∑k=1
limn→∞
∑(Ak • B
(ϕ)k
)n�
= limn→∞
K∑k=1
∑(Ak • B
(ϕ)k
)n�
= limn→∞
K∑k=1
ϕk((Ak)n�)
= limn→∞
K∑k=1
ϕ(s((Ak)n� ; k)) = limn→∞
ϕ
(K∑
k=1
s((Ak)n� ; k)
)= lim
n→∞ϕ((An�)K]
)= lim
n→∞ϕ((AK])n�
). (§)
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25
Next, let ρϕ = ϕ−ψϕ. We will show that ρϕ ∈ (L rκ )⊥. To see this, let A ∈ L r
κ . Then
AK] ∈ L rκ , and thus
∥∥∥∣∣∣(AK]
)n�
− AK]
∣∣∣∥∥∥r→ 0 for all K. Whence, by (§) and the
continuity of ϕ, we get ψϕ
(AK]
)= lim
n→∞ϕ((
AK]
)n�
)= ϕ
(AK]
)for all K. Since by
Corollary 3.2.3, we have∥∥∣∣AK] − A
∣∣∥∥r→ 0, it follows from the continuity of ψϕ and
ϕ that ψϕ(A) = ψϕ
(lim
K→∞AK]
)= lim
K→∞ψϕ
(AK]
)= lim
K→∞ϕ(AK]
)= ϕ
(lim
K→∞AK]
)=
ϕ (A). Hence ψϕ = ϕ on L rκ , which implies that ρϕ ∈ (L r
κ )⊥. Put P = {ψϕ : ϕ ∈(L r)∗}. We claim that (L r)∗ = P ⊕ (L r
κ )⊥ and P is isometrically isomorphic to(L r
κ )∗. From the definition of P, we have already had that (L r)∗ = P + (L rκ )⊥.
The decomposition (L r)∗ = P ⊕ (L rκ )⊥ will be obtained once it can be shown that
P ∩ (L rκ )⊥ = {0}. To see this, let ψϕ ∈ P ∩ (L r
κ )⊥ for some ϕ ∈ (L r)∗. Thenfor every A = {Ak}∞k=1 ∈ L r, we have by the absolute convergence of the series∞∑
k=1
∑Ak • B
(ϕ)k and the fact that the sequence An� ∈ L r
κ for all n that ψϕ(A) =
limn→∞
ψϕ (An�) = 0 . Therefore, ψϕ = 0, which yields P ∩ (L rκ )⊥ = {0}. Accordingly,
we have (L r)∗ = P⊕ (L rκ )⊥ as asserted. The rest is to prove hat P is isometrically
isomorphic to (L rκ )∗. To get this, we need to show first that
∥∥ψϕ|L rκ
∥∥ = ‖ψϕ‖. Itis obvious that
∥∥ψϕ|L rκ
∥∥ ≤ ‖ψϕ‖. To have that∥∥ψϕ|L r
κ
∥∥ ≥ ‖ψϕ‖, let ε > 0 begiven. Then there is a sequence A = {Ak}∞k=1 ∈ L r such that ‖|A|‖r ≤ 1 and
‖ψϕ‖ < |ψϕ(A)|+ ε. Thus, by the absolute convergence of the series∞∑
k=1
∑Ak •B
(ϕ)k ,
there is a positive integer n such that ‖ψϕ‖ < |ψϕ (An�)| + ε <∥∥ψψ|L r
κ
∥∥ + ε for allε > 0. This implies that ‖ψϕ‖ ≤ ∥∥ψψ|L r
κ
∥∥, and hence we obtain∥∥ψϕ|L r
κ
∥∥ = ‖ψϕ‖as desired. From this result, the map ψϕ �→ ψϕ|L r
κis now an isometric isomorphism
from P into (L rκ )∗. To see that it is onto, let ϕ0 ∈ (L r
κ )∗. We then have by theHahn Banach extension theorem that ϕ0 can extend uniquely to a bounded linearfunctional ϕ on L r with ‖ϕ‖ = ‖ϕ0‖. Since ψϕ agrees with ϕ on L r
κ , it followsψϕ|L r
κ= ϕ0. Consequently, the map ψϕ �→ ψϕ|L r
κis an isometric isomorphism from
P onto (L rκ )∗.
(3). Let ϕ = ψϕ + ρϕ ∈ (L r)∗. It is apparent that ‖ϕ‖ ≤ ‖ψϕ‖ + ‖ρϕ‖. Wewant to show that the reverse inequality holds. To prove this, let ε > 0 be given.Then there is a sequence A = {Ak}∞k=1 ∈ L r with ‖|A|‖r ≤ 1 such that |ψϕ(A)| >
‖ψϕ‖− ε
3. From this, we have by the absolute convergence of the series
∞∑k=1
∑Ak•B(ϕ)
k
that there exists a positive integer N such that |ψϕ (AN�)| > ‖ψϕ‖ − ε
3. Let C =
(sgnψϕ (AN�))AN� . Then ‖|C|‖r = ‖|AN� |‖r ≤ ‖|A|‖r ≤ 1 and ψϕ(C) = |ψϕ (AN�)| >
‖ψϕ‖ − ε
3. Next, let D = {Dk}∞k=1 ∈ L r be such that ‖|D|‖r ≤ 1, ρϕ(D) > 0 and
ρϕ(D) > ‖ρϕ‖− ε
3. Then by the absolute convergence of the series
∞∑k=1
∑Dk•B(ϕ)
k , we
have ψϕ(Dn�) → 0. Whence there is a positive integer J > N such that |ψϕ(DJ�)| <ε
3. Since D − DJ� ∈ L r
κ , it follows that ρϕ(DJ�) = ρϕ(D). Let E = C + DJ� . Then
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สมดกลาง
26
E ∈ L r and by Theorem 1.1(4), we have ‖|E|‖r = max{‖|C|‖r , ‖|DJ� |‖r} ≤ 1. Thus
‖ϕ‖ ≥ |ϕ(E)| = |ψϕ(E) + ρϕ(E)|= |ψϕ(C) + ψϕ(DJ�) + ρϕ(C) + ρϕ(DJ�)|= |ψϕ(C) + ψϕ(DJ�) + ρϕ(D)|≥ ψϕ(C) + ρϕ(D) − |ψϕ(DJ�)|> ‖ψϕ‖ − ε
3+ ‖ρϕ‖ − ε
3− ε
3
= ‖ψϕ‖ + ‖ρϕ‖ − ε.
Since ε > 0 was given arbitrarily, it follows that ‖ψϕ‖ + ‖ρϕ‖ ≤ ‖ϕ‖. Hence theequation ‖ϕ‖ = ‖ψϕ‖ + ‖ρϕ‖ is obtained.
Remark 3.2.6. Since (L rκ )∗ is isometrically isomorphic to P, we may treat (L r
κ )∗
as a subspace of (L r)∗. Thus Theorem 3.2.5 can be symbolized analogously to Dixmier’stheorem as follows:
(L r)∗ = (L rκ )∗ ⊕1 (L r
κ )s . ส
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Chapter 4
Conclusion
Let M∞ be the set of all infinite complex matrices. For each 1 ≤ r < ∞,we define a class of sequences of infinite complex matrices L r as follows:
L r =
{{[a
(k)ji
]}∞
k=1⊆ M∞ :
[ ∞∑k=1
∣∣∣a(k)ji
∣∣∣r] ∈ B(l2)
},
and for any sequence{[
a(k)ji
]}∞
k=1in M∞, we define
∥∥∥∣∣∣{[a(k)ji
]}∞
k=1
∣∣∣∥∥∥r
=
⎧⎪⎪⎨⎪⎪⎩∥∥∥∥∥[ ∞∑
k=1
∣∣∣a(k)ji
∣∣∣r]∥∥∥∥∥1/r
if{[
a(k)ji
]}∞
k=1∈ L r,
∞ otherwise.
In this thesis, we study some elementary properties and provide some results on theduality of L r. The main goal is to decompose the dual space (L r)∗ of L r as an l1
direct-sum of its two closed subspaces by a way analogous to a beautiful theorem ofDixmier on decomposing the dual B(l2)∗ of B(l2). The following are the results.
We first obtain some characterizations of the sets L r.
Theorem 4.1. Let {Ak}∞k=1 be a sequence in M∞ and 1 ≤ r < ∞. Then thefollowing are equivalent:
(1) {Ak}∞k=1 belongs to L r;
(2) Ak ∈ Sr2,2(C) for all k and the sequence
{n∑
k=1
A[r]k
}∞
n=1
is bounded in B(l2);
(3) the sequence
{∥∥∥∥∥n∑
k=1
A[r]k
∥∥∥∥∥}∞
n=1
is bounded.
To obtain that ‖|·|‖r is precisely a norm on L r, the following Holder-typeinequality is constructed.
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Theorem 4.2. (Holder-type inequality for sequences of matrices) For any sequences{Ak}∞k=1 and {Bk}∞k=1 in M∞,
‖|{Ak • Bk}∞k=1|‖1 ≤ ‖|{Ak}∞k=1|‖r ‖|{Bk}∞k=1|‖r∗ ,
where 1 < r < ∞ with1
r+
1
r∗= 1, under the conventions that ∞ · 0 = 0 · ∞ = 0,
∞ · α = α · ∞ = ∞ for all positive real number α and ∞ ·∞ = ∞.
From the Holder-type inequality, the corresponding Minkowski’s inequality isobtained.
Theorem 4.3. (Minkowski-type inequality for sequences of matrices) For any se-quences {Ak}∞k=1 and {Bk}∞k=1 in M∞ and 1 ≤ r < ∞,
‖|{Ak + Bk}∞k=1|‖r ≤ ‖|{Ak}∞k=1|‖r + ‖|{Bk}∞k=1|‖r ,
under the conventions that ∞ + α = α + ∞ = ∞ for all non-negative real number αand ∞ + ∞ = ∞.
From the Minkowski’s inequality, we have that the set L r equipped with thenorm ‖|·|‖r is a normed space. A Rieze-fischer-type theorem for completeness of thesequence spaces L r is obtained below.
Theorem 4.4. For each 1 ≤ r < ∞, the set L r equipped with the norm ‖|·|‖r is aBanach space.
For the classical sequence spaces lp (1 ≤ p < ∞), there is a result closely
related to their duality stating that for every 1 ≤ p < ∞ with1
p+
1
q= 1, a sequence
{xk}∞k=1 belongs to lq if and only if {xkyk}∞k=1 ∈ l1 for all {yk} ∈ lp. We obtain asimilar duality-type result for the sequence spaces L r as follows.
Theorem 4.5. Let 1 < r < ∞ with1
r+
1
r∗= 1.
(1) A sequence {Ak}∞k=1 ∈ L r∗ if and only if {Ak •Bk}∞k=1 ∈ L 1 for all {Bk}∞k=1 ∈L r.
(2) If {Ak}∞k=1 ∈ L r∗, then
‖|{Ak}∞k=1|‖r∗ = sup{‖|{Ak • Bk}∞k=1|‖1 : {Bk}∞k=1 ∈ L r, ‖|{Bk}∞k=1|‖r ≤ 1}.
Next, we define the class
L rκ =
{{[a
(k)ji
]}∞
k=1⊆ M∞ :
[ ∞∑k=1
∣∣∣a(k)ji
∣∣∣r] ∈ K(l2)
},
We obtain a characterization of L rκ as follows.
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Theorem 4.6. Let {Ak}∞k=1 be a sequence in M∞ and 1 ≤ r < ∞. Then {Ak}∞k=1 ∈L r
κ if and only if A[r]k is compact for all k and the sequence
{n∑
k=1
A[r]k
}∞
k=1
converges
in B(l2).
Theorem 4.7. For each 1 ≤ r < ∞, the set L rκ is a Banach subspace of L r.
We finally obtain a decomposition theorem for the dual (L r)∗ of L r as follows.
Theorem 4.8. Let 1 ≤ r < ∞. Then the following hold.
(1) The annihilator (L rκ )⊥ of L r
κ is a non-trivial closed subspace of the dual (L r)∗
of L r.
(2) There is a subspace P of (L r)∗ such that P is isometrically isomorphic to(L r
κ )∗ and (L r)∗ = P ⊕ (L rκ )⊥ .
(3) For any f ∈ (L r)∗, the decomposition f = g+h, where g ∈ P and h ∈ (L rκ )⊥,
satisfies ‖f‖ = ‖g‖ + ‖h‖.
Notice that Theorem 4.8 can be symbolized analogously to Dixmier’s theoremas follows:
(L r)∗ = (L rκ )∗ ⊕1 (L r
κ )s .
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Bibliography
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BIOGRAPHY NAME Mr. Suchat Samphavat
ADDRESS 59/2 Moo 1 Tambol Kuggatin, Amphur Muang, Ratchaburi, 70000 INSTITUTION ATTENDED
2009 Bachelor of Science in Mathematics, Silpakorn University
2012 Master of Science in Mathematics, Silpakorn University ส
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